Properties

Label 2-1520-19.7-c1-0-18
Degree $2$
Conductor $1520$
Sign $0.717 + 0.696i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.579 − 1.00i)3-s + (−0.5 − 0.866i)5-s + 2.43·7-s + (0.827 − 1.43i)9-s + 5.75·11-s + (0.797 − 1.38i)13-s + (−0.579 + 1.00i)15-s + (2.99 + 5.18i)17-s + (−0.149 + 4.35i)19-s + (−1.41 − 2.44i)21-s + (−0.470 + 0.814i)23-s + (−0.499 + 0.866i)25-s − 5.39·27-s + (−1.30 + 2.26i)29-s + 5.26·31-s + ⋯
L(s)  = 1  + (−0.334 − 0.579i)3-s + (−0.223 − 0.387i)5-s + 0.920·7-s + (0.275 − 0.477i)9-s + 1.73·11-s + (0.221 − 0.383i)13-s + (−0.149 + 0.259i)15-s + (0.725 + 1.25i)17-s + (−0.0342 + 0.999i)19-s + (−0.308 − 0.533i)21-s + (−0.0980 + 0.169i)23-s + (−0.0999 + 0.173i)25-s − 1.03·27-s + (−0.243 + 0.421i)29-s + 0.946·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.930464451\)
\(L(\frac12)\) \(\approx\) \(1.930464451\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.149 - 4.35i)T \)
good3 \( 1 + (0.579 + 1.00i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.43T + 7T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 + (-0.797 + 1.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.99 - 5.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.470 - 0.814i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.26T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 + (-3.15 - 5.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.26 - 3.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.47 + 7.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.09 + 1.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 + 9.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.26 + 9.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.504 + 0.874i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.41 - 7.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.12 - 8.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.80 - 6.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + (-5.55 + 9.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.02 + 3.51i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.354175967593545959854728087799, −8.331634094200924403784137180799, −7.951067044822817914151759161145, −6.82876737261001827181880342032, −6.19753603109699213714181540446, −5.36338250197367988725250795977, −4.13737228118847901971674097789, −3.61350813700661900686264140956, −1.65415628602824726986864591345, −1.13822038520596102667790051142, 1.14819192795897110374669973161, 2.47548948774566073190943349525, 3.85001691503578083364876978780, 4.48646726715338130337315725456, 5.26560428278391707787453623604, 6.34300825897482744508740731502, 7.18996403278469297808215106182, 7.85683744270320385464156373297, 9.024743387594016696553207981485, 9.430224958216998144943908790697

Graph of the $Z$-function along the critical line